Brittle-ductile transition stress of different rock types and its relationship with uniaxial compressive strength and Hoek–Brown material constant (mi)

Rocks deformed at low confining pressure are brittle, which means that after peak stress, the strength declines to a residual value established by sliding friction. The stress drop is the variation between peak and residual values. But no tension reduction takes place at high confining pressure. A proposed definition of the brittle-ductile transition is the transition pressure at which no loss in strength takes place. However, studies that consider information about the brittle-ductile transition, the criterion's range of applicability, how to determine mi, and how confining pressures affect mi's values are scarce. This paper aims to investigate the link between brittle-ductile transition stress, uniaxial compressive strength and Hoek–Brown material constant (mi) for different kinds of rock. It is essential to accurately determine the brittle-ductile transition stress to derive reliable values for mi. To achieve this purpose, a large amount of data from the literature was chosen, regression analysis was carried out, and brittle-ductile transition stress (σTR) was determined based on the combination of Hoek–Brown failure criteria and the recently used brittle-ductile transition stress limit of Mogi. Moreover, new nonlinear correlations were established between uniaxial compressive strength and Hoek–Brown material constant (mi) for different igneous, sedimentary and metamorphic rock types. Regression analyses show that the determination coefficient between σTR and UCS for gneiss is 0.9, sandstone is 0.8, and shale is 0.74. Similarly, the determination coefficient between σTR and mi for gneiss is 0.88. The correlation between Hoek–Brown material constant (mi) and σTR was not notable for sedimentary and metamorphic rocks, probably due to sedimentary rocks' stratification and metamorphic ones' foliation.


Theoretical background
Some carbonate rocks follow the A-type brittle-ductile transitions, particularly at high temperatures. In contrast, silicate rocks are considered to have B-type stress-strain curves (The typical stress-strain curves of A-type and B-type are schematically shown in Fig. 1a and b, respectively). Thus, the pressure dependence of the strength of rocks near the transition pressure is different between A-type and B-type. Most rocks, however, behave in an intermediate manner between A-type and B-type. An inelastic deformation occurs just before the transition pressure is reached, and after yielding, both fracturing and plastic deformation likely occurs. In addition, it was also suggested that a frictional sliding hypothesis applies to the brittle-ductile transition process of rocks (noted as B-type) in which the permanent deformation in the post-yield region occurs by cataclastic flow or frictional sliding 3 . Also, Kármán 20,21 published his measured failure limits as functions of the confining pressure. We had to read the data from the figures and recalculate them into MPa-they are collected in Tables 1 and 2, respectively.
With the increase of confining pressure, ductility, which is defined as the ability to undergo large permanent deformation without fracture, increases markedly, and a transition from the brittle to the ductile state takes place at some confining pressure 36 . Figure 3 shows the brittle-ductile behavior in the conventional triaxial compression test as a function of the confining pressure and compressive strength of silicate and carbonate rocks given by Mogi 2 . In silicate rocks, the brittle state region and the ductile state region are divided by a straight line passing through the origin (Fig. 2). This boundary line is expressed by (σ 1 − σ 3 ) = 3.4σ 3 .
In this section, to calculate the (σ TR ), the Mogi ductile-brittle transition stress equation and Hoek-Brown failure criteria are reformulated. The Hoek-Brown (H.B.) failure criterion is widely used in rock mechanics and rock engineering practice. This semi-empirical failure criterion was introduced by Hoek and Brown 40 , and the following form was suggested for intact rock 41 : where σ 1 and σ 3 are major and minor principal stress at failure, respectively, m i : Hoek-Brown material constant and σ c : the uniaxial compressive strength of intact rock. According to Eq. (1), two independent parameters are necessary, namely the: -Uniaxial compressive strength of the intact rock ( σ c ), -Hoek-Brown material constant of the intact rock (m i ).
(1) www.nature.com/scientificreports/ It should be noted that the Hoek-Brown criterion is proposed to deal with shear failure in rocks. Therefore, the Hoek-Brown criterion is only applicable for confining stresses within the range defined by σ 3 = 0 and the transition from shear to a ductile failure, as shown in Fig. 3. It was indicated that the range of σ 3 can significantly influence the calculation of m i 42,43 . Additionally, triaxial test data of Indiana limestone 44 shows that the applicability of the Hoek-Brown criterion is determined by the transition from shear to ductile failure at approximately σ 1 = 4σ 3 35 (Fig. 3).

Scientific Reports
Mogi 2 found that the average transition is defined as σ 1 = 4.4σ 3 , which is a convenient guide for selecting the maximum confining pressure for triaxial tests of intact rocks. Typical stress-strain curves in the brittle, the transition and the ductile state are very different (see Fig. 4). Brittle rocks break with a slight inelastic strain  www.nature.com/scientificreports/ and a rapid stress reduction after the peak stress, which is referred to as macroscopic failure 5 . The rock exhibits brittle-ductile transition behavior at moderate confining pressures, with a noticeable significant inelastic strain before reaching the peak stress, followed by a slow drop in stress 5,45 . When confining pressures are high, the rock becomes ductile, undergoing a substantial inelastic strain up to peak stress and remaining constant 46 . An empirical failure criterion has also been proposed; namely, for most rocks, the confining pressure must always be smaller than the uniaxial compressive strength to keep the brittle behavior of the rock 2 . Figure 5 illustrates the comparison of two criteria [Eqs. (2) and (3)] according to Zuo and Shen 48 . However, most experimental data in Fig. 6 shows that the brittle-ductile transition relationship may be nonlinear. The critical transition condition of brittle-ductile transition for rocks can be expressed by Eq. (2).  www.nature.com/scientificreports/ In Eq. (2), σ c = σ c σ t , µ is the friction coefficient, b is the fracture parameter of rocks. Equation (2) indicated that increasing σ c , the required σ 3 to initiate the σ TR increases. Figure 5 illustrates the comparison of two criteria [Eqs. (2) and (3)] 48 .
In this paper, based on the above listed analyses, the transition point from brittle to ductile failure is calculated using σ TR as referred to Mogi's widely used brittle-ductile transition limit for silicate rocks 2 : Thus: Substituting Eqs. (4a and 4b) with Eq. (1) we have the following equations:  www.nature.com/scientificreports/ σ 3 Can be derived from the following equation.   39 for m i value determination in silicate rocks and carbonate rocks, we have the Eqs. (9) and (10) for estimating σ TR , respectively.

Transition stress for different rock types
Through collecting the published data by Sheorey 49 , σ TR was calculated for different rock types. The data used in this paper is illustrated in Tables A.1, A.2 and A.3 for igneous, sedimentary, and metamorphic rocks, respectively (see Appendix). The correlations between σ TR and the UCS and m i are shown in Figs. 6, 7, and 8. As shown in Fig. 8, a high determination correlation was observed for sandstone, shale, and gneiss. (R 2 > 0.7); however, the correlation was weak for slate (R 2 < 0.5). Figure 6 shows that by increasing the values of m i and UCS, the values of σ TR increases; however, the amount of growth depends on the type of rock. For example, according to Fig. 7, for igneous rocks, as UCS increases, the values of σ TR increases with good data consistency and a high determination coefficient (R 2 = 0.89). Similarly, for sedimentary and metamorphic rocks, we can see good correlations; however, data consistency is not as significant as for igneous rocks. Figure 8 shows the comparison of the relationship between σ TR and published m i values. Specifically describing igneous rocks, we can see a strong correlation with a high determination coefficient (R 2 = 0.83), and good data consistency is notable. Since the constant of mi is an indicator of the brittleness of rock (50), the results show that the influence of m i on σ TR is more than UCS. All the empirical equations which derived from calculation and correlations are summarized in Table 3.

Discussion
According to our linear and nonlinear regression analyses for different rock types, Fig. 8 shows that σ TR calculated by this research has a high correlation with UCS in most types of rocks, and it can be used to estimate the transition stress of rocks based on their UCS. Figures 7 and 8 indicate that the best correlation was observed for igneous rocks and the reason is more probably related to the texture and the origin of the igneous rocks. A transition to ductile flow is predicted to occur when the strength as a function of pressure (or mean stress) has a slope that deviates from the relatively steep slope in the brittle faulting regime 2 . Implementing such criteria can be ambiguous since the "strength" in the ductile regime evolves with strain hardening and is not well defined. Accordingly, one has to arbitrarily assign it to be the stress attained at a fixed percentage of strain. Based on Eqs. (7) and (8), the value of σ TR is influenced non-linearly by the value of mi. In other words, as mi increases, σ TR increases. Equations (7) and (8) are in good agreement with the empirical failure criterion proposed by Mogi 2 which suggests that by increasing the rigidity of rock, the required confining pressure σ 3 that triggers brittle-ductile transition increases. In the same way, Tsikrikis et al. 50 performed a set of triaxial compressive tests on low-porosity carbonated rocks and observed that the σ TR decreases logarithmically with decreasing m i , increasing the average rock grain size and decreasing the ratio of the σ TR to the unconfined compressive strength σ tr σ c , but the stress ratio ( σ 1 σ 3 ) is approximately the same and independent of rock type, grain size, σ TR, and m i . Based on their analysis, it was found that σ TR can be formulated as a function of mi and σ c with the coefficient of determination of R 2 = 0.9, which shows good agreement with our research findings with respect to the relationship between m i and the ratio between transition stress and uniaxial compressive strength ( σ tr σ c ). Tsikrikis et al. 50 , conducted experimental tests over limestone and calculated m i = 23.5 and UCS = 66.6 MPa. Based on their measurements, the value of σ TR was 63 MPa. While using the proposed equation in this research [Eq. (8)] to calculate the σ TR, its value is 64 MPa which means that the results are close to each other (the data are summarized in Table A.4). On the other hand, for marble, they found the value of σ TR was 23.8 MPa; however, according to our formula, the value of σ TR is 28.3 MPa, which shows some discrepancies between the prediction of our model and their observation. Figure 9 shows the brittle, ductile, and brittle-ductile regions based on our proposed model [Eqs. (7) and (8)]. Compared with Hoek-Brown failure criteria, Mogi brittle-ductile  www.nature.com/scientificreports/ transition stress, and experimental data for granite, sandstone, and marble (see Table A  www.nature.com/scientificreports/ region is between 0 and σ 3 =41.34 MPa, the ductile region occurs at σ 3 > 41.34, and the brittle-ductile region occurs at σ TR = 41.34 MPa. The rocscience program 51 was used for the calculations. Similarly, Zuo and Shen 48 proposed a micromechanics-based frictional damage model to investigate the brittle-ductile transition process of various rocks and found that critical damage at failure can be linearly related to the level of confining pressure. The amount of ductile deformation and the strength increase progressively with increasing confining pressure until fully ductile deformation occurs with apparent work-hardening. This result can be linked to the micro-mechanics principle of m i conducted by Hoek and Martin 52 , which incorporates the role of coefficient of friction for pre-existing sliding crack surfaces and an intermediate fracture mechanics parameter that can be obtained from experimental data. They conclude that as the ratio of the coefficient of friction to the intermediate fracture parameter increase, the value of m i increases. Walton 29 analyzed the large database for different rock types. Based on his analysis, transition stress (σ TR ) depends on ductility parameter (d), UCS, and Hoek-Brown material constant (m i ). Through re-interpretation of previously published stress-strain data for a wide variety of rocks, silicate rocks (d) vary from 2.5 to 3, and for carbonate rocks is between 3.5 and 5, which is in good agreement with our results. Similarly, Iyare et al. 47 developed the experimental model based on a set of triaxial tests on mudstone samples to predict the σ TR They observed that for the tested samples, σ TR varies between 50 and 90 MPa, which is in good agreement with our proposed formula (Fig. 7) for the determination of σ TR based on UCS for sedimentary rocks with the coefficient of determination (R 2 = 0.7).
It is worth mentioning that more detailed material models beyond ideal elasticity give an exact relationship between rock strength parameters such as uniaxial compressive strength, m i, and σ TR Notably, the observed relations can be explained in a universal thermodynamic framework where internal variables characterise the structural changes 53,54 . These constitutive models are based only on universal principles of thermodynamics, are independent of particular mechanisms, and are successful in characterizing rheological phenomena in the brittle-ductile transition region of rocks, including and beyond simple creep and relaxation. This is in accordance with the difficulty of finding a very detailed quantitative mesoscopic mechanism for the brittle-ductile phenomena in rocks 55,56 .

Conclusions
This study represents an investigation of a large database of compression tests performed on different kinds of rocks over a wide range of confining stresses. The regression analyses of the relationships between uniaxial compressive strength (UCS), Hoek-Brown material constant (m i ), and brittle-ductile transition stress (σ TR ) showed that there is a new nonlinear correlation between uniaxial compressive strength and transition stress. This research reveals that the relation between the σ TR and UCS and m i is rock-type dependent. It means that for different rock types, the proposed formula has different material coefficients. For silicate rock (Granite), the slope of the brittle-ductile transition stress line, which we obtained in this research, is less than the slope of the brittle-ductile transition stress line for carbonate rocks (Marble). In other words, for silicate rock, the slope of the brittle-ductile transition line is shown by σ 1 = 4.4 σ 3 ; whereas for carbonated rocks in this research, the slope of the brittle-ductile transition line is shown by σ 1 = 6 σ 3 Regression analyses show that the determination coefficient between σ TR and UCS for gneiss is 0.9, sandstone is 0.8, and shale is 0.74. Similarly, the determination coefficient between σ TR and m i for gneiss is 0.88. Based on the regression analysis and due to the high determination coefficient between UCS and σ TR for different rocks, the UCS can be considered a significant parameter to estimate the σ TR . In addition, for the igneous rocks, both m i and UCS can be used for suggesting the σ TR . The result of this research can be used to estimate σ TR for different rock types in engineering practice. Future work should expand on the analyses presented in this paper, mainly focusing more on metamorphic rocks and considering the influences of fluid saturation and proper triaxial loading conditions on the brittle-ductile transition.

Data availability
The datasets generated and/or analyzed during the current study are available in the book of Sheorey 49 .